The Red Devils defender, who is the club's current longest-serving player, has recently completed his comeback from injury and is looking for regular first-team football.
He has made just one appearance for the club this term, which came in a 1-0 defeat to Wolves at Old Trafford earlier in January.
He impressed in that match, but with Harry Maguire and Raphael Varane the first-choice central defenders at the club, his game time will likely be limited under Ralf Rangnick.
According to L'Equipe, the 29-year-old is now a target for Bordeaux, who want to sign the former England international on loan.
The Ligue 1 club are on the hunt for a new defender and have had to shift players off their wage bill and squad to free up room to bring one in.
Defensive midfielder Otavio has gone to Atletico Miniero, while, English forward Josh Maja is heading to Stoke on loan, which includes an option to buy.
Jones has spent ten seasons at United since joining from Blackburn for £17million in 2011.
United have had a quiet window in terms of incoming transfers, but several players have departed the club.
Anthony Martial has left on loan for Sevilla, while Axel Tuanzebe and Amad Diallo have also found new temporary homes at Napoli and Rangers, respectively.
Donny van de Beek and Jesse Lingard could also be loaned out before the window slams shut on Monday.